# Ultraconnected space

From Topospaces

## Contents

## Definition

A topological space is termed an **ultraconnected space** if it satisfies the following equivalent conditions:

- It is nonempty and cannot be expressed as a union of two proper open subsets
- It is nonempty and cannot be expressed as a union of finitely many proper open subsets
- It is nonempty and any two nonempty closed subsets have nonempty intersection

## Relation with other properties

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

path-connected space | ultraconnected implies path-connected | |FULL LIST, MORE INFO | ||

connected space | |FULL LIST, MORE INFO | |||

normal space | ultraconnected implies normal | |FULL LIST, MORE INFO | ||

pseudocompact space | |FULL LIST, MORE INFO | |||

limit point-compact space | |FULL LIST, MORE INFO |

### Opposite properties

### Similar properties

- Irreducible space, with a similar definition but the roles of "open" and "closed" interchanged